Super-extreme event's influence on a Weierstrass-Mandelbrot Continuous-Time Random Walk

TL;DR

This paper investigates how super-extreme events, such as sustained drifts and shocks, significantly alter the velocity autocorrelation function in a generalized continuous-time random walk model, potentially enabling their detection in natural systems.

Contribution

It introduces a WM-CTRW framework to analyze the impact of super-extreme events on VAF, highlighting their distinct effects compared to typical extreme events.

Findings

Super-extreme events cause substantial changes in VAF.

Different types of super-extreme events produce distinct VAF signatures.

Alterations in VAF could allow detection of super-extreme events in real systems.

Abstract

Two utmost cases of super-extreme event's influence on the velocity autocorrelation function (VAF) were considered. The VAF itself was derived within the hierarchical Weierstrass-Mandelbrot Continuous-Time Random Walk (WM-CTRW) formalism, which is able to cover a broad spectrum of continuous-time random walks. Firstly, we studied a super-extreme event in a form of a sustained drift, whose duration time is much longer than that of any other event. Secondly, we considered a super-extreme event in the form of a shock with the size and velocity much larger than those corresponding to any other event. We found that the appearance of these super-extreme events substantially changes the results determined by extreme events (the so called "black swans") that are endogenous to the WM-CTRW process. For example, changes of the VAF in the latter case are in the form of some instability and distinctly differ from those caused in the former case. In each case these changes are quite different compared to the situation without super-extreme events suggesting the possibility to detect them in natural system if they occur.